Here is what I think is going on. Since the Owl did not include any context to establish a pronoun referent. "It" could be refering to any -n word or -t word. In this nebulous situation, you have to say something, so say DET. If the owl had given us a referent noun, and asked us to translate thusly: "She has the apple. Does she eat it? " you would reply : "Hun har aeblet. Spiser hun DET?" because you know that "it" refers to a -t noun. Similarly, If the Owl asked: "She has the cheese, does she eat it?" you would teply " Hun har osten. Spiser hun DEN?"

My take on this would be that the English word "does" does indeed translate to "gør" by itself, but when it comes to forming questions or negations, Danish doesn't use do-support (nor does any other language apart from English, at least in this form), i.e. you don't need the verb "do" to form these constructions.